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Paul
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What is the volatility smile?

January 1st, 2003, 5:59 pm

...and what causes it?!P
 
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Longkappa
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What is the volatility smile?

January 1st, 2003, 10:26 pm

Vol smile is the manifestation of non-lognormal distributions of return. It causes the BS model to give higher implied volatilities for OTM and ITM options. This reflects the fat tails or kurtosis of the actual distribution.
 
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plessas
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Joined: March 9th, 2002, 10:23 pm

What is the volatility smile?

January 2nd, 2003, 2:53 am

Substituting in the BS equation the market price for an option you can back out the implied volatility. Since you refer to the same underlying asset you expect that to be the same over different strikes. However in practise this is not always the case. Putting those implied volatilities along with the respective strike prices on YX axes you get a U-SHAPED curve (instead of a straight line, parallel to the strike axis) which is called a volatility smile. Sometimes it is possible to get the reverse pattern which is called a volatility frown. As LongKappa suggests volatility smile reflects the fat tails of the actual distribution when compared to the lognormal one that the BS model assumes for the returns.rgds,Dimitris
 
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tubul
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Joined: August 16th, 2002, 4:56 am

What is the volatility smile?

January 2nd, 2003, 7:36 am

From first principles, the value of a call option is the integral (0 to infinity) of Max (S - PV(X),0). Under the BS framework, the risk neutral distribution is lognormal. In reality of course it is not lognormal. A series of options on a stock (with different strikes) tells us something about the underlying risk neutral distribution. However they all give us information on the same distribution, so it is reasonable to expect that the second moment of the distribution should be the same, irrespective of which option we are looking at. That is indeed the case. Non-zero 3rd & 4th moments will imply possibilities of very large and small S values when evaluating the integral. These will clearly have a bearing on the way deep ITM & OTM options are priced. If one is able to evaluate the integral (for various strikes) and then find the volatilities that equates the BS eqn with the true option values, then one gets a smile.The vol smiles thus don't really reflect the true second moment of the distribution. They can be thought of as a 'fudge factor' to make the BS model produce reasonable option prices (and as a tool for helping us estimate the triue second moment itself). The true volatility (as per the risk neutral distribution) is of course independent of the strike.Interest rate and equity markets have frowns. The equity frown is generally thought to be caused by crashophobia - ie the market players assigning a non-zero probability of a very large downward move (this started happening after the 1987 crash). This causes the risk neutral distribution to get skewed and hence cause an asymmetry in the smile shape.Foreign exchange markets have vol curve shapes closer to smiles than frowns. That is because vols are sensitive to the perceived action of central banks over the life of the option and circumstances can cause an intervention by them leading to either depreciation or appreciation. That brings about a greater extent of symmetry in the smile. However, this may not hold true for emerging mkt currencies where a visible trend can be seen, indicting that the central bank is comfortable with a unidirectional move (and thus implying an asymmetry in the distribution).It is possible that the smile shape may also be influenced by supply/demand factors and hence differing liquidity in the options across various strikes. An ATM option is generally more liquid than an expensive deep ITM option. The difference in liquidity will lead to differing bid/ask spreads and not building in a correction for this will cause a change in the slope of the smile.Regards,Tubul
 
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reza
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What is the volatility smile?

January 2nd, 2003, 4:08 pm

How do we model the smile? by using Non Log-Normal distributions, namely1- Stochastic Volatility (two Brownian Motions)2- Jumps (Jump Diffusion, VG, CGMY and other Levy Processes)3- Local Vol (Implied distribution)what else??
 
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newton
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What is the volatility smile?

January 2nd, 2003, 6:32 pm

It appears that the vol smile is caused by (at least) two components fo the volatility. One, of course, is the volatility of the underlying (Gaussian), and the second is the volatility caused by the probability of a crash (Poisson). As you go more and more OTM, the volatility (option price) caused by the underlying volatility decreases, but the volatility caused by a chance of a crash remains constant. Put - Call parity causes the ITM Call to also shake a bit.
 
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Anthis
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Joined: October 22nd, 2001, 10:06 am

What is the volatility smile?

January 3rd, 2003, 11:34 am

It has been argued that there is an smile assymetry as well particularly for equity and bond options, that means the implied vol for a put is higher from the implied vol of a call for options of the same maturity and equivalent moneyness.Is the above statement true?If yes, is this due to higher demand for puts by institutional investors for hedging needs as well as existence of short sales restrictions (regulatory or not) and the well documented assymetry between buy and sell execution costs in those underlying markets? RegardsAnthis
 
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kr
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Joined: September 27th, 2002, 1:19 pm

What is the volatility smile?

January 3rd, 2003, 3:30 pm

Smile can also be partly explained by the 'leverage effect' - that the equity becomes riskier when the price goes lower. In particular, in order to make stocks go to zero - which is observed - the vol cannot remain finite as S --> 0. I would not argue that that completely explains smile, but it does explain why we see 'smirk' more than 'smile' these days.
 
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Johnny
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What is the volatility smile?

January 3rd, 2003, 3:40 pm

In addition to KR's point, it's worth pointing out that options on single shares can behave quite differently to those on equity indices. Typically the former have more of a fixed vol surface (corresponding to a relatively stable capital structure) whilst the latter tend to have more of a floating vol surface.
 
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filthy

What is the volatility smile?

January 3rd, 2003, 3:46 pm

another reason for the higher implied vol of downside strikes is simply that volatilitytends to rise as the underlying price drops. a conjectured reason for this is that traders who will not dynamically hedge their optionstend to buy puts (as insurance) and sell calls (covered writes). in addition to the obvious supply/demandissue this creates for the market makers we have the situation that when the market drops the shortput player (who is dynamically hedging) has to sell into the breaking market, thus exacerbating its decline.so dynamic hedging raises downside vol.a similar effect damps upside vol.maybe...this is all very tough to verify.
 
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mghiggins
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Joined: November 3rd, 2001, 1:38 pm

What is the volatility smile?

January 4th, 2003, 12:54 pm

Qualitative explanation for why there is a vol smile in stochastic volatility models from a market-making perspective:First, let's step back and look at the standard Black-Scholes world. Why do options have value over intrinsic value? Because they have positive Gamma. As a market-making trader, you buy an option and Delta-hedge it - now, the value of your portfolio as a function of the underlying asset price is parabolic, and you make money whichever way the asset price moves. Nice! You're willing to pay for that position, and the more volatile the asset price is (so the higher you expect to shoot up the sides of the parabolic payoff), the more you're willing to pay for the option.Now let's imagine that volatility is stochastic (but we'll ignore the possibility of jumps in the asset price).A few definitions first (these aren't standard terms, but I like them better than other labels I've heard before, like Vanna - yeck, that sounds like bad Indian food): Vega: sensitivity of my portfolio value to moves in volatility. d(Value)/d(Vol). All vanilla options have positive Vega.Vega Gamma: sensitivity of Vega to moves in vol. d(Vega)/d(Vol) or d^2(Value)/d(Vol)^2. Out-of-the-money options, both high- and low-strike, have positive Vega Gamma; ATM options have roughly zero Vega Gamma.Vega DSpot: sensitivity of Vega to moves in asset price (spot). d(Vega)/d(Spot) or d^2(Vega)/d(Vol)/d(Spot). High-strike options have positive Vega DSpot; low-strike options have negative Vega DSpot; ATM options have roughly zero Vega DSpot.Why there's a smile:Imagine you buy an option with positive Vega Gamma (any OTM option, either high- or low-strike). You hedge the outright Vega by selling an ATM option (which has roughly zero Vega Gamma). Now you're got a parabolic payoff vs volatility, much like you did vs asset price in the Gamma example at the top. So, whichever way vol moves, you make money. Woo hoo! You're willing to pay up for that portfolio, above the standard zero-stochastic-vol (Black-Scholes) value. The more volatile vol is, the more you're willing to pay.So this means that, if volatility is stochastic, you tend to be willing to pay more than the Black-Scholes value for OTM options, but not for ATM options (since they have no Vega Gamma). This means that implied volatilities will be higher for OTM options than ATM options, which is the smile.Why there's a skew:Imagine you buy an option with positive Vega DSpot (a high-strike option) and hedge the outright Vega with an ATM option. Also imagine there's a positive correlation between moves in spot (the asset price) and vol. If spot moves up, your Vega turns positive (since d(Vega)/d(Spot) is positive), but because there's a positive correlation between moves in spot and vol, you expect vol to go up too. Vol is going up when Vega is positive, so you make money. If spot goes down your Vega turns negative, but you expect vol to drop as well because of the positive correlation. So you make money there too. Whichever way spot moves you make money - and again you're willing to pay up for that portfolio. The more positive the correlation is, and the more volatile vol is, the more you're willing to pay.In this example, if the spot/vol correlation had been negative, you'd lose money whichever way spot moved, and you would have to be paid (relative to the zero-correlation price) to take on the portfolio. Similarly if Vega DSpot were negative (low-strike options) and correlation were positive.So this drives the skew: if the spot/vol correlation is positive, you're willing to pay more for high-strike options and less for low-strike options that the zero-correlation price. This gives a positive skew. If the spot/vol correlation is negative, the reverse is true, and the correlation leads to a negative skew.This is all a bit approximate, since the discussion has been focussed on the instantaneous Vega Gamma and Vega DSpot - and to get an accurate price for a derivative you have to include the global behaviour. However, the local behaviour often drives the pricing because the spot is most likely to stay near where it started.
 
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WaaghBakri
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What is the volatility smile?

January 12th, 2003, 11:11 pm

Vega DSpot: sensitivity of Vega to moves in asset price (spot). d(Vega)/d(Spot) or d^2(Vega)/d(Vol)/d(Spot). Typo: d^2(Value)/d(Vol)/d(Spot) ?
 
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mghiggins
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Joined: November 3rd, 2001, 1:38 pm

What is the volatility smile?

January 13th, 2003, 11:40 am

Thanks! Nice catch. Though it won't let me edit my message to fix the mistake now.
 
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WaaghBakri
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Joined: March 21st, 2002, 4:07 am

What is the volatility smile?

January 13th, 2003, 5:57 pm

Intuitively, as the time to expiry increases one imagines that the asset terminal distribution (actual or model) is more diffused - smaller peak & fatter tails. One would also expect that as the time to expiry increases, the percentage difference of the "energy" in the fat tails between the actual & lognormal gets smaller & smaller. Therefore one would expect that for long-dated options the volatility smile is less pronounced (ie flatter). Is it?It would be great if someone could also describe a typical evolution of a vol. smile from its birth to expiry, ie how does the smile change with the passage of time. Many thanks for all your posts.
 
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Johnny
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Joined: October 18th, 2001, 3:26 pm

What is the volatility smile?

January 14th, 2003, 8:07 am

WBTypically the smile evolves just as you described. Pretty flat for long dated options; pretty steep for short dated options. Smile gets steeper as the expiry date approaches.